\contentsline {section}{\numberline {1}Introduction}{3}{section.1}
\contentsline {section}{\numberline {2}All-on to All-off is Realizable}{3}{section.2}
\contentsline {subsection}{\numberline {2.1}Induction Proof by Graph Theory}{3}{subsection.2.1}
\contentsline {subsection}{\numberline {2.2}Algebra proof by Linear Algebra}{4}{subsection.2.2}
\contentsline {paragraph}{Interpretation of Gaussian elimination on Graph}{5}{section*.2}
\contentsline {subsection}{\numberline {2.3}Historical Review}{6}{subsection.2.3}
\contentsline {section}{\numberline {3}Completely Solvable}{6}{section.3}
\contentsline {subsection}{\numberline {3.1}Grid Graph}{6}{subsection.3.1}
\contentsline {subsubsection}{\numberline {3.1.1}$\sigma $-game}{7}{subsubsection.3.1.1}
\contentsline {subsubsection}{\numberline {3.1.2}$\sigma ^+$-game}{10}{subsubsection.3.1.2}
\contentsline {subsection}{\numberline {3.2}Several APR Graph Classes }{11}{subsection.3.2}
\contentsline {subsubsection}{\numberline {3.2.1}Path}{12}{subsubsection.3.2.1}
\contentsline {subsubsection}{\numberline {3.2.2}Spider}{13}{subsubsection.3.2.2}
\contentsline {subsubsection}{\numberline {3.2.3}Caterpillars}{13}{subsubsection.3.2.3}
\contentsline {subsubsection}{\numberline {3.2.4}APR Trees}{14}{subsubsection.3.2.4}
\contentsline {subsection}{\numberline {3.3}Historical Reviews}{15}{subsection.3.3}
\contentsline {section}{\numberline {4}Optimize Problems}{15}{section.4}
\contentsline {subsection}{\numberline {4.1}Minimize Odd Parity Set}{15}{subsection.4.1}
\contentsline {subsubsection}{\numberline {4.1.1}$\mathbf {NP}$-Complete on General Graph}{16}{subsubsection.4.1.1}
\contentsline {subsubsection}{\numberline {4.1.2}Minimal Activation Set for Several Graph Classes}{16}{subsubsection.4.1.2}
\contentsline {paragraph}{Path}{17}{section*.3}
\contentsline {paragraph}{Cycle}{17}{section*.4}
\contentsline {paragraph}{Complete Bipartite Graph}{18}{section*.5}
\contentsline {paragraph}{Series-Parallel Graph}{18}{section*.6}
\contentsline {subsection}{\numberline {4.2}Bounds for Off State}{20}{subsection.4.2}
\contentsline {subsubsection}{\numberline {4.2.1}Maximizing Off Switches(MOS)}{20}{subsubsection.4.2.1}
\contentsline {paragraph}{Hard Approximation of MOS}{22}{section*.7}
\contentsline {subsubsection}{\numberline {4.2.2}Fixed-Parameter Problem}{23}{subsubsection.4.2.2}
\contentsline {subsubsection}{\numberline {4.2.3}Lower Bounds for Tree}{24}{subsubsection.4.2.3}
\contentsline {subsection}{\numberline {4.3}Historical Review}{26}{subsection.4.3}
